Question

Find the mean deviation about the median for the following data: 4, 6, 9, 3, 10, 13, 2.

Answer 1

Hint: As we know, median of ungrouped data is the middle term of the sorted data given in the question. Also, the median of the ungrouped data is different for even and odd numbers of terms. Firstly, we need to observe the data whether it has an even or odd number of terms then we will find the middle term of the sorted data. For an odd number of terms, the middle term can simply be calculated by $\dfrac{{n + 1}}{2}$ and this term will be the median. For even number of terms, there will be two middle terms which is calculated by $\dfrac{n}{2},\dfrac{n}{2} + 1$ and then these two terms can be added and divided by 2 to get the median of data. The mean deviation about the median is calculated by $\dfrac{{\sum {\left| {{x_i} - M} \right|} }}{n}$ . Now, put the value of median in the formula and mean deviation about median is obtained.

Complete step-by-step answer:
The ungrouped data is given in the question.
The given data is 4, 6, 9, 3, 10, 13, 2.
It is not sorted. So, we will first sort this data in ascending order which will be:
2, 3, 4, 6, 9, 10, 13
To calculate the median of the ungrouped data, we have to observe the number of terms given.
There are 7 terms which is odd.
So, the median can be calculated by finding the middle term.
The middle term of the given data is:
$\dfrac{{n + 1}}{2}$
Here, we have 7 numbers of terms which means $n = 7$.
So, the middle term will be 4.
The 4th term is 6.
Therefore, the median of the given data is 6.
Now, we have to calculate mean deviation about the median.
Mean deviation about median is given by $\dfrac{{\sum {\left| {{x_i} - M} \right|} }}{n}$
On putting values we get,
\[
  \dfrac{{\sum {\left| {{x_i} - M} \right|} }}{n} = \dfrac{{\sum {\left| {{x_i} - 6} \right|} }}{7} \\
   = \dfrac{{\left| {2 - 6} \right| + \left| {3 - 6} \right| + \left| {4 - 6} \right| + \left| {6 - 6} \right| + \left| {9 - 6} \right| + \left| {10 - 6} \right| + \left| {13 - 6} \right|}}{7} \\
   = \dfrac{{23}}{7} = 3.28 \\
 \]
Therefore, mean deviation about median is 3.28.

Hence, median is 6 and mean deviation about median is 3.28.

Note: The calculation of median and mean deviation about median is simple but it should also be rechecked to avoid any mistakes in these types of questions. Arithmetic Mean in the most common and easily understood measure of central tendency.